Fags of 4chan

The suffix -fag probably started as a bad attempt at an insult somewhere back in the dark days of history. Soon it became so widespread and applicable to any form of belief, subculture, activity and so on, that it became obsolete. Today, the suffix -fag is just a slangism that carries no fixed positive or negative connotation by itself, and is used to describe someone simply as belonging to a certain group of people. As far as meaning is concerned, it could be replaced by something like "-person" (newfag -> new person). Whether a certain *-fag is used as an insult or not is instead derived from: a) the context of the discussion, especially the person using the word and; b) the part preceding the suffix and not the suffix itself. It can even be used as a sign of affection.

It has been noted that due to the suffix of -fag on 4chan can be as demeaning or endearing as the word "child" is to normal people. Plus implications of man-sex.

For example, a person from Europe might refer to himself as a Eurofag, with a neutral tone, simply indicating that he hails from Europe. "Yeah, I'm a Eurofag, so I can't watch American football on my TV."

A different (probably non-European) person might use it as an insult. In this case it is the "euro-" part that serves as the carrier of all known negative stereotypes and connotations about Europeans, by the insulting person's cultural viewpoint (e.g., "Shut up Eurofag, go suck some Muslim cock.")

A third usage is almost the same as the first example, but with a more positive connotation. "Eurofags make the best music."

Someone who attributes a name to their posts on 4chan. Namefagging is generally considered a terrible thing, which can and will lead to a 300+ post debate about the merits and disadvantages of identity on imageboards. Basically, don't do it unless you need to be distinguished from other anons.

Might be called "namefags+." Tripcodes are like little keys, to show people that the other guy posting as BobsUrUnkle isn't really you. Much like namefagging, you should avoid this unless using it to establish identity for a certain project (such as Weaver when running RubyQuest). No other purpose is acceptable.

Touhoufags are fans of the Touhou video game series, which you can ask /jp/ about. You can recognize them by the image of little girls in frilly dresses they add to each message they post, sometimes weirdly relevant to the context of the thread.
"Touhoufag" is a specific user on /tg/ with encyclopedic knowledge of D&D 3.X, D&D 4e, and WoD. He was banned for a long time by Nazi Mod, but returned around the release of D&D Next. An army of Touhoufags (possibly including that one) are making D&D 4th edition power cards using Touhou fanart ripped from Danbooru.

A softer form of identification on threads, in which an anon uploads various images of the same character along each of their posts. Avatarfags are not usually hated with as much unadulterated vehemence of their Namefag and Tripfag bretheren, but they are still mocked by the superior anons, usually with implications that the neckbeard responsible has a crush on his avatar.

Any one person related in any way, shape or form to the (whatever) in the title. You can put anything there, seriously. E.g.: "Eurofag," "Amerifag," "Mathfag," "4fag," "straightfag," "WoWfag," "FagFagFaggityFag-Fag".
The term for a "Japanfag" is Weeaboo.
Common types of fags who are seen frequenting /tg/ are:

Homosexual. Sometimes used as a generic insult, though usually with little effect. Can be referred to as a Fagfag or Gayfag.

Faggot is also an archaic term for a bundle of sticks, specifically the pile of sticks that witches were burned over for being homosexual, most likely where 'fag' as colloquial language for a cigarette or a bassoon came from. This is also were the term "Flaming" came from.

It is consequently hilarious to read The Lord of the Rings whenever they talk about "throwing faggots [the bundle of stick kind] onto the fire".